Embedded newform 43.7.d.a.7.7

• Label: 43.7.d.a.7.7
• Level: $43$
• Weight: $7$
• Character: 43.7
• Analytic conductor: $9.892$
• Analytic rank: $0$
• Dimension: $42$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	43.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.89232559565\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Relative dimension:	\(21\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			7.7
Character	\(\chi\)	\(=\)	43.7
Dual form			43.7.d.a.37.15

$q$-expansion

\(f(q)\)	\(=\)	\(q-6.95456i q^{2} +(-14.3583 + 8.28978i) q^{3} +15.6341 q^{4} +(9.86801 - 5.69730i) q^{5} +(57.6517 + 99.8557i) q^{6} +(-310.875 - 179.484i) q^{7} -553.820i q^{8} +(-227.059 + 393.278i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 3 q^{3} - 1194 q^{4} - 3 q^{5} + 111 q^{6} - 147 q^{7} + 3852 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	6.95456i		−	0.869320i	−0.900595	−	0.434660i	\(-0.856869\pi\)
							0.900595	−	0.434660i	\(-0.143131\pi\)
\(3\)	−14.3583	+	8.28978i	−0.531789	+	0.307029i	−0.741745	−	0.670682i	\(-0.766001\pi\)
							0.209955	+	0.977711i	\(0.432668\pi\)
\(5\)	9.86801	−	5.69730i	0.0789441	−	0.0455784i	−0.460008	−	0.887915i	\(-0.652154\pi\)
							0.538952	+	0.842336i	\(0.318820\pi\)
\(7\)	−310.875	−	179.484i	−0.906342	−	0.523277i	−0.0270899	−	0.999633i	\(-0.508624\pi\)
							−0.879253	+	0.476356i	\(0.841957\pi\)
\(11\)	−1005.14			−0.755176			−0.377588	−	0.925974i	\(-0.623246\pi\)
							−0.377588	+	0.925974i	\(0.623246\pi\)
\(13\)	−901.804	+	1561.97i	−0.410471	+	0.710956i	−0.994941	−	0.100459i	\(-0.967969\pi\)
							0.584471	+	0.811415i	\(0.301302\pi\)
\(17\)	−1332.58	+	2308.09i	−0.271235	+	0.469793i	−0.969178	−	0.246360i	\(-0.920765\pi\)
							0.697943	+	0.716153i	\(0.254099\pi\)
\(19\)	−2771.47	+	1600.11i	−0.404064	+	0.233286i	−0.688236	−	0.725487i	\(-0.741615\pi\)
							0.284172	+	0.958773i	\(0.408281\pi\)
\(23\)	−516.012	−	893.760i	−0.0424108	−	0.0734577i	0.844041	−	0.536279i	\(-0.180170\pi\)
							−0.886452	+	0.462821i	\(0.846837\pi\)
\(29\)	−27655.6	−	15967.0i	−1.13394	−	0.654680i	−0.189017	−	0.981974i	\(-0.560530\pi\)
							−0.944923	+	0.327294i	\(0.893863\pi\)
\(31\)	−12598.6	−	21821.3i	−0.422898	−	0.732481i	0.573324	−	0.819329i	\(-0.305654\pi\)
							−0.996222	+	0.0868483i	\(0.972320\pi\)
\(37\)	13186.5	−	7613.22i	0.260330	−	0.150301i	−0.364155	−	0.931338i	\(-0.618642\pi\)
							0.624485	+	0.781037i	\(0.285309\pi\)
\(41\)	34456.6			0.499943			0.249972	−	0.968253i	\(-0.419579\pi\)
							0.249972	+	0.968253i	\(0.419579\pi\)
\(43\)	60437.9	−	51658.8i	0.760158	−	0.649738i
							\(47\)	101043.			0.973226			0.486613	−	0.873618i	\(-0.338232\pi\)
0.486613	+	0.873618i	\(0.338232\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.d.a.7.7	✓	42
43.37	odd	6	inner	43.7.d.a.37.15	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.d.a.7.7	✓	42	1.1	even	1	trivial
43.7.d.a.37.15	yes	42	43.37	odd	6	inner